Graph y=sin(x-(3pi)/2)

Problem

y=sin(x−(3*π)/2)

Solution

1. Identify the parent function and its key properties. The parent function is y=sin(x) which has a period of 2*π an amplitude of 1 and starts at the origin (0,0) moving upwards.

2. Determine the phase shift by looking at the horizontal displacement. The expression x−(3*π)/2 indicates a horizontal shift to the right by (3*π)/2 units.

3. Calculate key points for one period by adding the phase shift to the standard xcoordinates of y=sin(x)

• Start: 0+(3*π)/2=(3*π)/2

• Quarter point: π/2+(3*π)/2=2*π

• Midpoint: π+(3*π)/2=(5*π)/2

• Three-quarter point: (3*π)/2+(3*π)/2=3*π

• End: 2*π+(3*π)/2=(7*π)/2

4. Apply the cofunction identity to simplify the expression for easier graphing. Note that sin(x−(3*π)/2) is equivalent to cos(x)

sin(x−(3*π)/2)=sin(x)*cos((3*π)/2)−cos(x)*sin((3*π)/2)

sin(x−(3*π)/2)=sin(x)*(0)−cos(x)*(−1)

sin(x−(3*π)/2)=cos(x)

5. Plot the points and draw the smooth wave. The graph is a cosine wave starting at (0,1) with intercepts at (π/2,0) and ((3*π)/2,0) and a minimum at (π,−1)

Final Answer

y=sin(x−(3*π)/2)=cos(x)

Want more problems? Check here!